What is the first thing that comes to mind when you hear the word "patterns?" For most people, they think of art and design. But there are patterns all around us, even if we don't see them at first.
Throughout this project, you will be working with a series of virtual manipulatives, all while exploring patterns you may have never seen before.
As you work through the various manipulatives, some guiding questions will be given to you. Some questions you will have to develop on your own. In the end, you will be doing the work of a mathematician.
Task
Throughout this project, you will be asked to work with a series of virtual manipulatives. You may find them quite enjoyable. For each manipulative you work with, there will be a series of questions you will have to think about as you work through them. Keep the questions in mind, as they will you to finding the patterns that are present. Make sure you keep a record of your answers to the guiding questions.
Above is a photomosaic of Yoda, where each part of the picture is made up of a frame from the Star Wars movies. Each by itself is a single picture, but put together we have Yoda! The closer you get to the poster, the better you can see the individual frames. The further you get from the poster, the more it looks like Yoda.
Process
Utah State University has created a website known as the National Library of Virtual Manipulatives (NLVM). This site allows anyone with internet access the ability to explore math by using manipulatives, where users get a hands-on experience with various mathematical concepts. Within the site, you will find manipulatives that help show how balancing an equation works, how to convert units, or even how relationships exist in functions. All of the manipulatives contain instructions to help you better understand how each one works.
*IT IS HIGHLY RECOMMENDED THAT YOU READ THE INSTRUCTIONS FOR EACH MANIPULATIVE!*
For those of you who may have played the video game Star Wars: Knights of the Old Republic or worked on solving this puzzle in my room or Mr. Zakula's room, you will recognize this manipulative.
What is the minimum number of moves necessary to complete the Towers when there are 4 discs? 5 discs? 6? 7? 8?
Make a table where the independent variable is the number of discs and the dependent variable is the minimum number of moves. What pattern develops?
Examining your table and pattern, can you predict how many moves are needed for 10 discs? 20?
How do you know which move to make based on the number of discs you have to move?
In this manipulative, you need to figure out how to place numbers in overlapping circles so that the sum of each circle is 0. You will need to work with a few different sets of Circle 0 before you realize a pattern.
Once you get the hang of Circle 0, you will probably notice that you are making a particular move to begin. What move are you trying to start with? Why do you begin with that move?
Here is a puzzle that has been around for ages, often done with frogs instead of pegs. The idea is to have all of the pegs switch spots, where each peg can only move one spot forward into an empty spot or jump over one peg of a different color. Try to determine a pattern with the four peg game, then work your way up to the eight peg game. Keep in mind that as you solve the puzzle, you need to work on what the pattern is.
What should your first move be?
Is there an algorithm you can write that will work on any number of pegs?
You may have already played this game, which uses the skills of logic. You can set it to use anywhere from two to six colors. The more colors there are, the more difficult it gets. Start out working with the two-color game and keep track of how many moves it takes you to figure out the solution, as well as how you are formulating your guesses. Keep working on the two-color game until you can solve it quickly and easily, then advance to three colors, then four colors, working your way up to working with all six colors.
What was the guess you were making for your first move to start the game? Does it stay the same when you add more colors?
How did you eliminate different colors?
This game works a lot like a proof, as you should base your next move from what you learned from previous moves. Try to write a rule as to how to move from one step to the next.
This is an exercise in deduction. When using deduction, you are trying to find a pattern from your observations, which happend to be what youv'e been doing up to this point. Here, you have to try and find a counterfeit coin that has a slightly different weight than the other coins. You need to figure out an efficient process to find the counterfeit coin. To do this, you want to use the fewest number of weighings as possible.
What process did you use to find the counterfeit coin? Did you compare it with another proess to see if it was the most efficient?
What was the fewest number of times you needed to weigh 8 coins? 9 coins? 12 coins?
For the trials done with 8 and 9 coins, the manipulative asks you to solve it in two weighings. Do you think it is possible? How can it be done?
This is a classic probability scenario. The idea behind this manipulative was used on a game show known as "Let's Make a Deal!" Play through some rounds to see what kinds of prizes you can win. The manipulative will keep track of your wins and losses, as well as the percentages for each
Try a few different sets of 50 games where you choose to "stick" with the same door every time. What do you notice about the percentage for winning?
Now tray sets of 50 games where you choose to "switch." What do you notice about the percentage of winning now?
Now just guess whatever you want: stick or switch, it doesn't matter! Just do sets of 50 games using no predetermined pattern. Now what is the percentage of winning?
Do you think the show's producers were giving away too many prizes by playing by these rules? Why do you think so?
Here is another situation that showed up in Star Wars: Knights of the Old Republic. For this activity, you need to fill and empty cylinders so that a certain amount of fluid is left inside. There is a definite pattern that comes up (you might even find more than one pattern). Can you figure it out?
Which cylinder are you using more? The larger or smaller one?
How would this skill be useful in the real world?
Evaluation
Category
4
3
2
1
Mathematical Concepts
Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).
Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s).
Explanation shows some understanding of the mathematical concepts used to solve the problem(s).
Explanation shows very limited understanding of the mathematical concepts used to solve the problem(s) OR is not written.
Mathematical Reasoning
Uses complex and refined mathematical reasoning.
Uses effective mathematical reasoning.
Some evidence of mathematical reasoning.
Little or no evidence of mathematical reasoning.
Use of Manipulatives
Student always listens and follows directions and only uses manipulatives as instructed.
Student typically listens and follows directions and only uses manipulatives as instructed most of the time.
Student sometimes listens and follows directions and only uses manipulatives appropriately when reminded.
Student rarely listens and often "plays" with manipulatives instead of using them as instructed.
Explanation
Explanation is detailed and clear.
Explanation is clear.
Explanation is a little difficult to understand and is missing several critical components.
Explanation is difficult to understand and is missing several critical components OR was not included.
Neatness and Organization
The work is presented in a neat, clear, organized fashion that is easy to follow.
The work is presented in a neat and organized fashion that is usually easy to follow.
The work is presented in an organized fashion but may be difficult to follow at times.
The work appeears sloppy and unorganized. It is difficult to know what information goes together.
Strategy and Procedures
Typically uses an efficient and effective strategy to solve the problem(s).
Typically uses an effective strategy to solve the problem(s).
Sometimes uses an effective strategy to solve the problem(s), but does not do so consistently.
Rarely uses an effective strategy to solve problem(s).
Completion
All problems are completed.
All but one of the problems are completed.
All but two of the problems are completed.
Several of the problems are not completed.
Conclusion
By going through the virtual manipulatives, you should have determined that there were patterns that allowed us to understand the different concepts. This is how math was discovered. People wondered how things worked, so they examined the different situations that arose. They realized that there were patterns that governed over math and devised algorithms to explain them. By this point, you should be well on your way to becoming a mathematician like the following men:
The Great Gauss
Pierre de Fermat
René Descartes
Blaise Pascal
But wait, mathematicians (as well as people in other professions) have to report their findings, and so do you! You will choose one of the math manipulatives and create a presentation on what you discovered. You will present the following information:
How did you develop the pattern you used to solve the problem given with the manipulative?
How could the pattern be applied to the real world?
How does the pattern work?
Be creative with you findings. You could write a news report. You could created a PowerPoint or Keynote presentation. You could create a poster, or make a movie, or use photos and Comic Life. Use the questions from the process to help guide you. Make sure to include your answers to those questions with your findings. You will meet with me to discuss your project as you are creating it, and then present it to the entire class.
Resources
Click on the links to visit the original sources Yoda Photomosaic NLVM RubiStar for creating a rubric Pictures of mathematicians and mathematician information:
It is important to note that all computers that will be using this project have updated Flash and Shockwave. If a computer does not have the required software, a prompt will show up that will guide you to the proper page to update or receive the plug-in.
This project was originally designed for an Algebra 2 class to complete. I have used it primarily with my Math for Standards class, and it is usable for grades 9-12. The project may initially be met with resistance at first if students are not used to completing a project of this type. For students who are used to these types of projects, they should be quite receptive. Many of my students have enjoyed this project after having worked with it for a few days. Students do enjoy being able to work in a hands-on manner with the math.
For the WebQuest, the amount of time that is given to work with the manipulatives will vary based on the skill level of the students working on them. Be flexible as you determine the amount of time that will be given to work online. Time should also be given for students to put together their presentations, and a day in class should be given for individual feedback with each student/group. A day or two should also be set aside for presentations.
As mentioned in the process, it is important for the students to read the instructions for each manipulative. You may want to walk them through how to work the manipulatives, as well, without giving away the patterns. You can also use this as an explanation of how mathematics came to be discovered and how this thought process can be implemented with any type of problem, not just math.
Standards
Math Standards:
2.2.11.A. - Develop and use computation concepts, operations and procedures with real numbers in problem-solving situations.
2.4.11.A. - Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.
2.4.11.B. - Construct valid arguments from stated facts.
2.4.11.C. - Determine the validity of an argument.
2.4.11.E. - Demonstrate mathematical solutions to problems.
2.5.11.A. - Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.
2.5.11.C. - Present mathematical procedures and results clearly, systematically, succinctly, and correctly.
2.5.11.D. - Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.
2.6.11.D. - Determine the validity of the sampling method described in a given study.
2.6.11.H. - Use sampling techniques to draw inferences about large populations.
2.7.11.B. - Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.
2.8.11.A. - Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.
2.8.11.B. - Give examples of patterns that occur in data from other disciplines.
2.8.11.C. - Use patterns, sequences and series to solve routine and non-routine problems.
Science and Technology Standards:
3.2.10.B. - Apply process knowledge and organize scientific and technological phenomena in varied ways.
3.2.10.C. - Apply the elements of scientific inquiry to solve problems.
3.7.10.C. - Apply basic computer operations and concepts.
3.7.10.D. - Utilize computer software to solve specific problems.
Reading, Writing, Speaking, and Listening Standards:
1.2.11.A. - Read and understand essential content of informational texts and documents in all academic areas.
1.4.11.B. - Write complex informational pieces (e.g., research papers, analyses, evaluations, essays).
1.5.11.A. - Write with a sharp, distinct focus.
1.5.11.B. - Write using well-developed content appropriate for the topic.
1.5.11.C. - Write with controlled and/or subtle organization.
1.5.11.A. - Listen to others.
1.6.11.C. - Speak using skills appropriate to formal speech situations.
Table of Contents
Introduction
What is the first thing that comes to mind when you hear the word "patterns?" For most people, they think of art and design. But there are patterns all around us, even if we don't see them at first.Throughout this project, you will be working with a series of virtual manipulatives, all while exploring patterns you may have never seen before.
As you work through the various manipulatives, some guiding questions will be given to you. Some questions you will have to develop on your own. In the end, you will be doing the work of a mathematician.
Task
Throughout this project, you will be asked to work with a series of virtual manipulatives. You may find them quite enjoyable. For each manipulative you work with, there will be a series of questions you will have to think about as you work through them. Keep the questions in mind, as they will you to finding the patterns that are present. Make sure you keep a record of your answers to the guiding questions.Process
Utah State University has created a website known as the National Library of Virtual Manipulatives (NLVM). This site allows anyone with internet access the ability to explore math by using manipulatives, where users get a hands-on experience with various mathematical concepts. Within the site, you will find manipulatives that help show how balancing an equation works, how to convert units, or even how relationships exist in functions. All of the manipulatives contain instructions to help you better understand how each one works.*IT IS HIGHLY RECOMMENDED THAT YOU READ THE INSTRUCTIONS FOR EACH MANIPULATIVE!*
Towers of Hanoi
Click for manipulativeFor those of you who may have played the video game Star Wars: Knights of the Old Republic or worked on solving this puzzle in my room or Mr. Zakula's room, you will recognize this manipulative.
Circle 0
Click for manipulativeIn this manipulative, you need to figure out how to place numbers in overlapping circles so that the sum of each circle is 0. You will need to work with a few different sets of Circle 0 before you realize a pattern.
Think you're good at Circle 0? Try these out!
Peg Puzzle
Click here for manipulativeHere is a puzzle that has been around for ages, often done with frogs instead of pegs. The idea is to have all of the pegs switch spots, where each peg can only move one spot forward into an empty spot or jump over one peg of a different color. Try to determine a pattern with the four peg game, then work your way up to the eight peg game. Keep in mind that as you solve the puzzle, you need to work on what the pattern is.
Mastermind
Click here for manipulativeYou may have already played this game, which uses the skills of logic. You can set it to use anywhere from two to six colors. The more colors there are, the more difficult it gets. Start out working with the two-color game and keep track of how many moves it takes you to figure out the solution, as well as how you are formulating your guesses. Keep working on the two-color game until you can solve it quickly and easily, then advance to three colors, then four colors, working your way up to working with all six colors.
Coin Problem
Click here for manipulativeThis is an exercise in deduction. When using deduction, you are trying to find a pattern from your observations, which happend to be what youv'e been doing up to this point. Here, you have to try and find a counterfeit coin that has a slightly different weight than the other coins. You need to figure out an efficient process to find the counterfeit coin. To do this, you want to use the fewest number of weighings as possible.
Stick or Switch
Click here for manipulativeThis is a classic probability scenario. The idea behind this manipulative was used on a game show known as "Let's Make a Deal!" Play through some rounds to see what kinds of prizes you can win. The manipulative will keep track of your wins and losses, as well as the percentages for each
Fill and Pour
Click here for manipulativeHere is another situation that showed up in Star Wars: Knights of the Old Republic. For this activity, you need to fill and empty cylinders so that a certain amount of fluid is left inside. There is a definite pattern that comes up (you might even find more than one pattern). Can you figure it out?
Evaluation
Conclusion
By going through the virtual manipulatives, you should have determined that there were patterns that allowed us to understand the different concepts. This is how math was discovered. People wondered how things worked, so they examined the different situations that arose. They realized that there were patterns that governed over math and devised algorithms to explain them. By this point, you should be well on your way to becoming a mathematician like the following men:But wait, mathematicians (as well as people in other professions) have to report their findings, and so do you! You will choose one of the math manipulatives and create a presentation on what you discovered. You will present the following information:
- How did you develop the pattern you used to solve the problem given with the manipulative?
- How could the pattern be applied to the real world?
- How does the pattern work?
Be creative with you findings. You could write a news report. You could created a PowerPoint or Keynote presentation. You could create a poster, or make a movie, or use photos and Comic Life. Use the questions from the process to help guide you. Make sure to include your answers to those questions with your findings. You will meet with me to discuss your project as you are creating it, and then present it to the entire class.Resources
Click on the links to visit the original sourcesYoda
Photomosaic
NLVM
RubiStar for creating a rubric
Pictures of mathematicians and mathematician information:
- Descartes site; picture
- Fermat site; picture
- Gauss site; picture
- Pascal site; picture
It is important to note that all computers that will be using this project have updated Flash and Shockwave. If a computer does not have the required software, a prompt will show up that will guide you to the proper page to update or receive the plug-in.This project was originally designed for an Algebra 2 class to complete. I have used it primarily with my Math for Standards class, and it is usable for grades 9-12. The project may initially be met with resistance at first if students are not used to completing a project of this type. For students who are used to these types of projects, they should be quite receptive. Many of my students have enjoyed this project after having worked with it for a few days. Students do enjoy being able to work in a hands-on manner with the math.
For the WebQuest, the amount of time that is given to work with the manipulatives will vary based on the skill level of the students working on them. Be flexible as you determine the amount of time that will be given to work online. Time should also be given for students to put together their presentations, and a day in class should be given for individual feedback with each student/group. A day or two should also be set aside for presentations.
As mentioned in the process, it is important for the students to read the instructions for each manipulative. You may want to walk them through how to work the manipulatives, as well, without giving away the patterns. You can also use this as an explanation of how mathematics came to be discovered and how this thought process can be implemented with any type of problem, not just math.
Standards
Math Standards:2.2.11.A. - Develop and use computation concepts, operations and procedures with real numbers in problem-solving situations.
2.4.11.A. - Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.
2.4.11.B. - Construct valid arguments from stated facts.
2.4.11.C. - Determine the validity of an argument.
2.4.11.E. - Demonstrate mathematical solutions to problems.
2.5.11.A. - Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.
2.5.11.C. - Present mathematical procedures and results clearly, systematically, succinctly, and correctly.
2.5.11.D. - Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.
2.6.11.D. - Determine the validity of the sampling method described in a given study.
2.6.11.H. - Use sampling techniques to draw inferences about large populations.
2.7.11.B. - Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.
2.8.11.A. - Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.
2.8.11.B. - Give examples of patterns that occur in data from other disciplines.
2.8.11.C. - Use patterns, sequences and series to solve routine and non-routine problems.
Science and Technology Standards:
3.2.10.B. - Apply process knowledge and organize scientific and technological phenomena in varied ways.
3.2.10.C. - Apply the elements of scientific inquiry to solve problems.
3.7.10.C. - Apply basic computer operations and concepts.
3.7.10.D. - Utilize computer software to solve specific problems.
Reading, Writing, Speaking, and Listening Standards:
1.2.11.A. - Read and understand essential content of informational texts and documents in all academic areas.
1.4.11.B. - Write complex informational pieces (e.g., research papers, analyses, evaluations, essays).
1.5.11.A. - Write with a sharp, distinct focus.
1.5.11.B. - Write using well-developed content appropriate for the topic.
1.5.11.C. - Write with controlled and/or subtle organization.
1.5.11.A. - Listen to others.
1.6.11.C. - Speak using skills appropriate to formal speech situations.